In a business context, how does the derivative of the log-sum-exp function relate to the exponential function in terms of optimizing a portfolio's expected return?

In a portfolio problem we often write the log‑sum‑exp of the weighted returns as \( \ell(\mathbf{w})=\log\! \big(\sum_i e^{r_i w_i}\big) \); its derivative with respect to a weight \(w_j\) is \(\partial \ell/\partial w_j = \frac{e^{r_j w_j}}{\sum_i e^{r_i w_i}}\), which is exactly the softmax of the weighted returns. This shows that the marginal benefit of increasing a weight is proportional to the exponential of that asset’s return, so assets with higher returns get larger incremental gains. When we maximize expected return, the gradient points us toward allocating more to assets whose exponentials are larger, i. e.